Chapter 4 – Finite Fields

Introduction

• will now introduce finite fields

• of increasing importance in cryptography– AES, Elliptic Curve, IDEA, Public Key

• concern operations on “numbers”– what constitutes a “number”– the type of operations and the properties

• start with concepts of groups, rings, fields from abstract algebra

Group

• a set of elements or “numbers”– A generalization of usual arithmetic

• obeys: – closure: a.b also in G – associative law: (a.b).c = a.(b.c) – has identity e: e.a = a.e = a – has inverses a-1: a.a-1 = e

• if commutative a.b = b.a – then forms an abelian group

• Examples in P.105

Cyclic Group

• define exponentiation as repeated application of operator– example: a3 = a.a.a

• and let identity be: e=a0

• a group is cyclic if every element is a power of some fixed element– ie b = ak for some a and every b in group

• a is said to be a generator of the group• Example: positive numbers with addition

Ring

• a set of “numbers” with two operations (addition and multiplication) which are:

• an abelian group with addition operation • multiplication:

– has closure– is associative– distributive over addition: a(b+c) = ab + ac

• In essence, a ring is a set in which we can do addition, subtraction [a – b = a + (–b)], and multiplication without leaving the set.

• With respect to addition and multiplication, the set of all n-square matrices over the real numbers form a ring.

Ring

• if multiplication operation is commutative, it forms a commutative ring

• if multiplication operation has an identity element and no zero divisors (ab=0 means either a=0 or b=0), it forms an integral domain

• The set of Integers with usual + and x is an integral domain

Field

• a set of numbers with two operations:– Addition and multiplication– F is an integral domain– F has multiplicative reverse

• For each a in F other than 0, there is an element b such that ab=ba=1

• In essence, a field is a set in which we can do addition, subtraction, multiplication, and division without leaving the set. – Division is defined with the following rule: a/b = a (b–1)

• Examples of fields: rational numbers, real numbers, complex numbers. Integers are NOT a field.

Definitions

Modular Arithmetic

• define modulo operator a mod n to be remainder when a is divided by n– e.g. 1 = 7 mod 3 ; 4 = 9 mod 5

• use the term congruence for: a ≡ b (mod n) – when divided by n, a & b have same remainder – eg. 100 ≡ 34 (mod 11)

• b is called the residue of a mod n– since with integers can always write: a = qn + b

• usually have 0 <= b <= n-1 -12 mod 7 = -5 mod 7 = 2 mod 7 = 9 mod 7

Modulo 7 Example

... -21 -20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 ... all numbers in a column are equivalent (have same

remainder) and are called a residue class

Divisors

• say a non-zero number b divides a if for some m have a=mb (a,b,m all integers) – 0 ≡ a mod b

• that is b divides into a with no remainder

• denote this b|a

• and say that b is a divisor of a

• eg. all of 1,2,3,4,6,8,12,24 divide 24

Modular Arithmetic Operations

• has a finite number of values, and loops back from either end

• modular arithmetic– Can perform addition & multiplication – Do modulo to reduce the answer to the finite

set

• can do reduction at any point, ie– a+b mod n = a mod n + b mod n

Modular Arithmetic

• can do modular arithmetic with any group of integers: Zn = {0, 1, … , n-1}

• form a commutative ring for addition

• with an additive identity (Table 4.2)

• some additional properties– if (a+b)≡(a+c) mod n then b≡c mod n– but (ab)≡(ac) mod n then b≡c mod n

only if a is relatively prime to n

Modulo 8 Example

Greatest Common Divisor (GCD)

• a common problem in number theory

• GCD (a,b) of a and b is the largest number that divides both a and b – eg GCD(60,24) = 12

• often want no common factors (except 1) and hence numbers are relatively prime– eg GCD(8,15) = 1– hence 8 & 15 are relatively prime

Euclid's GCD Algorithm

• an efficient way to find the GCD(a,b)• uses theorem that:

– GCD(a,b) = GCD(b, a mod b)

• Euclid's Algorithm to compute GCD(a,b): – A=a, B=b– while B>0

•R = A mod B•A = B, B = R

– return A

Example GCD(1970,1066)1970 = 1 x 1066 + 904 gcd(1066, 904)1066 = 1 x 904 + 162 gcd(904, 162)904 = 5 x 162 + 94 gcd(162, 94)162 = 1 x 94 + 68 gcd(94, 68)94 = 1 x 68 + 26 gcd(68, 26)68 = 2 x 26 + 16 gcd(26, 16)26 = 1 x 16 + 10 gcd(16, 10)16 = 1 x 10 + 6 gcd(10, 6)10 = 1 x 6 + 4 gcd(6, 4)6 = 1 x 4 + 2 gcd(4, 2)4 = 2 x 2 + 0 gcd(2, 0)• Compute successive instances of GCD(a,b) = GCD(b,a mod b).• Note this MUST always terminate since will eventually get a mod b

= 0 (ie no remainder left).

Galois Fields

• finite fields play a key role in many cryptography algorithms

• can show number of elements in any finite field must be a power of a prime number pn

• known as Galois fields• denoted GF(pn)• in particular often use the fields:

– GF(p)– GF(2n)

Galois Fields GF(p)

• GF(p) is the set of integers {0,1, … , p-1} with arithmetic operations modulo prime p

• these form a finite field– since have multiplicative inverses

• hence arithmetic is “well-behaved” and can do addition, subtraction, multiplication, and division without leaving the field GF(p)– Division depends on the existence of multiplicative

inverses. Why p has to be prime?

Example GF(7)

Example: 3/2=5GP(6) does not exist

Finding Inverses

• Finding inverses for large P is a problem• can extend Euclid’s algorithm:

EXTENDED EUCLID(m, b)1. (A1, A2, A3)=(1, 0, m);

(B1, B2, B3)=(0, 1, b)2. if B3 = 0

return A3 = gcd(m, b); no inverse3. if B3 = 1

return B3 = gcd(m, b); B2 = b–1 mod m4. Q = A3 div B35. (T1, T2, T3)=(A1 – Q B1, A2 – Q B2, A3 – Q B3)6. (A1, A2, A3)=(B1, B2, B3)7. (B1, B2, B3)=(T1, T2, T3)8. goto 2

Inverse of 550 in GF(1759)

Prove correctness

Polynomial Arithmetic

• can compute using polynomials• several alternatives available

– ordinary polynomial arithmetic– poly arithmetic with coefficients mod p– poly arithmetic with coefficients mod p and

polynomials mod another polynomial M(x)

• Motivation: use polynomials to model Shift and XOR operations

Ordinary Polynomial Arithmetic

• add or subtract corresponding coefficients

• multiply all terms by each other

• eg– let f(x) = x3 + x2 + 2 and g(x) = x2 – x + 1

f(x) + g(x) = x3 + 2x2 – x + 3

f(x) – g(x) = x3 + x + 1

f(x) x g(x) = x5 + 3x2 – 2x + 2

Polynomial Arithmetic with Modulo Coefficients

• when computing value of each coefficient, modulo some value

• could be modulo any prime

• but we are most interested in mod 2– ie all coefficients are 0 or 1– eg. let f(x) = x3 + x2 and g(x) = x2 + x + 1

f(x) + g(x) = x3 + x + 1

f(x) x g(x) = x5 + x2

Modular Polynomial Arithmetic

• Given any polynomials f,g, can write in the form:– f(x) = q(x) g(x) + r(x)– can interpret r(x) as being a remainder– r(x) = f(x) mod g(x)

• if have no remainder say g(x) divides f(x)• if g(x) has no divisors other than itself & 1 say it

is irreducible (or prime) polynomial• Modular polynomial arithmetic modulo an

irreducible polynomial forms a field– Check the definition of a field

Polynomial GCD

• can find greatest common divisor for polys• GCD: the one with the greatest degree

– c(x) = GCD(a(x), b(x)) if c(x) is the poly of greatest degree which divides both a(x), b(x)

– can adapt Euclid’s Algorithm to find it:– EUCLID[a(x), b(x)]1. A(x) = a(x); B(x) = b(x)2. 2. if B(x) = 0 return A(x) = gcd[a(x), b(x)]3. R(x) = A(x) mod B(x)4. A(x) ¨ B(x)5. B(x) ¨ R(x)6. goto 2

Modular Polynomial Arithmetic

• can compute in field GF(2n) – polynomials with coefficients modulo 2– whose degree is less than n– Coefficients always modulo 2 in an operation– hence must modulo an irreducible polynomial

of degree n (for multiplication only)

• form a finite field• can always find an inverse

– can extend Euclid’s Inverse algorithm to find

Example GF(23)

Computational Considerations

• since coefficients are 0 or 1, can represent any such polynomial as a bit string

• addition becomes XOR of these bit strings

• multiplication is shift & XOR– Example in P.133

• modulo reduction done by repeatedly substituting highest power with remainder of irreducible poly (also shift & XOR)

Summary

• have considered:– concept of groups, rings, fields– modular arithmetic with integers– Euclid’s algorithm for GCD– finite fields GF(p)– polynomial arithmetic in general and in GF(2n)